Nature/nurture. As children develop they are shaped by two factors, their genetics and their enviornment. The mental health of the parent would impact both. Genetically many mental health disorers are generationally passed on. The stability of a parent greatly affects the child developmentally. Different theories of development identify different levels of impact. Overall, most theories agree childhood is.
Much of psych. The relationship between childrens' well-being and that of their parents has been studied since antiquity. The famous ancient greek physician galen was tutored by his father, whom he called " a man of justice, modesty, and goodness." much of modern child developmental psychology examines this relationship as well. But children are much more than mere copies of their parents!
To a great degree. Parents are their children's role model. Parenting is a tough job , oftentimes we get caught off guard when confronted w situations our children present.Likewise children are put in the same scenario.Both over-react, get out of control. When we are calm, patient, compassionate, positive attitude always, they will grow up emotionally stable, healthier & can cope w the world, instead of being defiant.
Observation Impt. At least 90% of what kids learn comes from watching what others in their lives do, how others act, how they relate.... Considering that kids spend most of (at least) their early yrs w/their caretakers (often parents), it's no surprise that those w/ anxious parent(s) tend 2worry about "what if's" (what if something bad happens); kids of depressed parent(s) can get withdrawn, pessimistic; etc.
Parenting. Parenting is tough, even when all members of the family are " neurotypical." if mental illness impacts a parent's ability to be responsive, engaged and able to set clear, consistent limits with age-appropriate expectations, that parent should receive help without shame or blame, especially since " the apple doesn't fall far from the tree.".
It depends. On the diagnosis and the success of the treatment.